Plasma assisted modified betatron

ABSTRACT

In a modified Betatron a low density background plasma is maintained in the vacuum chamber causing image charges in response thereto to form in the chamber wall. These image charges cause the self forces of the electron beam being accelerated in the betatron to be directed inward in the polodial plane thus eliminating injection problems, the diamagnetic to paramagnetic transition, and the l=2 resistive wall instability.

BACKGROUND OF THE INVENTION

The present invention relates to high current, high energy electronbeams. More specifically, the present invention relates to a modifiedbetatron which has beam self forces that are inward in the polodialplane.

There is a great interest in the development of high current, highenergy electron beams. The principal technique to accelerate electronrings, which comprise the electron beam, is a conventional betatron.See, for instance U.S. Pat. No. 4,392,111, by Rostaker, 1983. However,this technique is limited by the fact that the outward self forces ofthe electron beam must be smaller than the externally imposed focusingforces of the betatron. These focusing forces are controlled by thegradient of the vertical(z) field, the field parallel to the axis of thebetatron. The focusing is parameterized by the field index ##EQU1## For0<η<1, the focusing forces are inward in both the radial and verticaldirections in the poloidal plane. In practice, this limits the currentto several hundred amps. Another scheme is to use a plasma betatron.See, Taggert et al., "Successful Betatron Acceleration of KiloampereElectron Rings in RECE-Christa", Physical Review Letters, Vol. 52, No.18, p. 1601-1604, April 1984, for a more complete description of aplasma betatron. Here a runaway tokamak discharge is produced which canhave a current of many kiloamps and a voltage of 10 MeV or more.However, the beam has a large energy spread with no way to extract itfrom either the plasma or the magnetic field. Another scheme is amodified betatron which uses a toroidal magnetic field to overcome thelack of external field focusing. See U.S. Pat. No. 4,481,475, byKapetanakos and Sprangle, 1984, for a more detailed description of themodified betatron.

However, the disadvantages of the modified betatron are threefold. Firstof all, one must inject the beam across toroidal field lines in order tohave a beam centered in the liner of the betatron. The current scheme ofKapetanakos et al., Phys, Rev. Lett. 49, 741 (1982) proposes to shootthe beam into the toroidal vacuum chamber near the liner. The drift dueto the focusing fields and image fields causes the beam to drift in thepoloidal plane around the liner. In one toroidal transit (about 20 nsec)it should drift enough to miss the injector. In one poloidal drift time(several hundred nanoseconds), external fields can be changed to bringthe beam slightly in from the liner so that it misses the injector againand henceforth. On a longer time scale, wall resistivity causes the beam(if the current is sufficiently low) to drift inward. This occursbecause the liner has finite conductivity. The effect of this finiteconductivity is to cause a drag force on the beam which causes the beamto spiral either in or out, depending on the beam current.

However there is a significant range of beam currents for which wallresistivity causes the beam to drift outward if it is near the liner,but inward if it is near the center. Since it is unlikely that the beamcan reverse drift directions on the way in, the injection scheme ofKapetanakos et al., supra. appears to be viable only for fairly low beamcurrents. For higher beam currents still, the beam will drift outward nomatter what its position is in the poloidal plane.

A second possible difficulty concerning the modified betatron is thediamagnetic to paramagnetic transition. See W. M. Manheiner and J. M.Finn, Particle Accel. 14, 29 (1983); J. M. Finn and W. M. Manheimer,Phys. Fluids 26, 3400 (1983). Depending on whether the net self force isoutward or inward in the poloidal plane, the net electron drift velocityin the poloidal plane is in the diamagnetic or paramagnetic direction. Adiamagnetic drift velocity means that the electron poloidal velocitycrossed with the toroidal magnetic field (right hand rule) produces aninward force in the poloidal plane. A paramagnetic drift means that thisforce is outward in the poloidal plane. At low energy, where the outwardself fields are stronger than the focusing forces, the beam must have anadditional inward force to maintain equilibrium. At high energy, wherethe self forces are weaker than the focusing forces, the beam must havean additional outward force to maintain equilibrium. These forces areprovided by the poloidal (diamagnetic or paramagnetic) drift velocitytimes the toroidal magnetic field. Thus as the high current beamaccelerates, it makes a transition from diamagnetic to paramagneticcurrent flow. It has been shown that subject only to the constraint thatthe acceleration time τ_(a) ≈10⁻³ sec is very long compared to the drifttime, τ_(D) ≧10⁻⁷ sec, this transition must suddenly change the topologyof the beam orbits in the poloidal plane. Whether the beam can survivesuch a sudden, violent perturbation is an open question.

Finally, although the focusing fields in the modified betatron stabilizethe l=1 resistive wall instability, l=2 modes are still unstable andpose a real threat to beam confinement in the modified betatron. See R.G. Kleva, E. Ott and P. Sprangle, Phys. Fluids 26, 2689 (1983). The l=1modes causes a displacement of the beam center in the poloidal plane.The l=2 modes causes a distortion of the shape of the beam in thepoloidal plane from circular to elliptical.

The three fundamental issues identified regarding the high currentmodified betatron operating with a vacuum background: beam injection,the diamagnetic to paramagnetic transition, and the l=2 resistive wallinstability will now be more fully discussed.

One of the important issues for the modified betatron is injecting thebeam. The present thinking for the Naval Research Laboratory modifiedbetatron experiment is described in Kapetanakos, et al., supra. The beamis injected near the liner and drifts around the edge of the linerthrough a combination of drift paths generated by the focusing fields(field index) of the betatron and image forces in the vacuum chamberwall due to the beam. The former is directed inward in the poloidalplane, the latter, outward. At high beam current the latter dominatesand the beam rotates in a counterclockwise direction as in FIG. 3 ofKapetanakos et al., supra. for the case of a ten kilo Amp beam. If thecombination of forces is large enough, the drift velocity will be greatenough so that after one toroidal revolution, the beam will be displacedin the poloidal plane by a large enough distance that it misses theinjector. Then, since it has many more toroidal transits before it wouldhit the injector again, macroscopic fields could change sufficiently tobring the beam into the center.

One potential problem with this scheme, is that for higher currentbeams, the net poloidal force on the beam is outward near the liner, butinward when the beam is at the center. Thus, as the beam continues tospiral in the poloidal plane, at some point it must reverse direction.To see this more quantitatively, if the field index of the beam is 1/2,which gives rise to optimal confinement in the radial and verticaldirection, the focusing field produces an inward poloidal force on acharge q of ##EQU2## where R_(o) is the major radius of the equilibriumorbit, B_(z) is the vertical field and ρ is the displacement of the beamfrom the equilibrium orbit in the poloidal plane ρ² =(R-R_(o))² +z². Theimage electric force for a cylindrical system is given by ##EQU3## wherea is the minor radius of the liner.

The actual force is canceled in part by magnetic forces and also by anyfractional neutralization f. The fractional neutralization describes thefact that positive ions, with a density of f times the beam densitycould also be confined by the beam. If the beam is near the wall (ρ≦a)the net poloidal drift velocity is given by ##EQU4## where δ is thedistance from the beam center to the liner and γ is the electron energydivided by the electron rest energy. Since the beam enters the toroidalliner right near the outer edge of the liner, δ is roughly equal to thebeam radius ρ_(b).

If the beam is near the center (so ρ<a), the poloidal drift is given by##EQU5## For the case of a vacuum modified betatron, the current can beclassified as being in one of three ranges:

I. High Current ##EQU6## II. Intermediate Current ##EQU7## III. LowCurrent ##EQU8##

In the high current regime, the forces on the beam in the poloidal planeare outward and the beam always rotates in the counterclockwisedirection. In the intermediate regime, the forces are outward when thebeam is near the wall, but inward when the beam is near the center. Thusin this current regime, the beam must reverse its direction of rotationbefore it gets to the center. Also, at some radius between the centerand the wall, the beam will have zero poloidal drift velocity. It seemslikely that some time after injection, an intermediate current beam willstagnate around this point and gradually fill the chamber. In the lowcurrent regime, the inward focusing forces always dominate and the beamrotation is clockwise.

It is also worth noting that if the liner is resistive, the beam willspiral inward if the net force is inward and visa versa. Thus aresistive wall can only trap a class III low current beam. It ispossible that an intermediate current beam can be trapped if it can bebrought sufficiently near the center that the net forces are inward. Forthe parameters of the NRL modified betatron experiment, (see Sprangle etal., supra; Kapentanakas et al., supra, B.sub.θ =2×10³, γ=4, R_(o) =10²,a=15, δ=2, B_(z) =140, the lower and upper currents of the intermediaterange are 3.2×10³ A and 1.2×10⁴ A. Thus the maximum current which can betrapped by wall resistivity in a vacuum modified betatron is about 3 kA.Actually however, the maximum current is less because at the 3.2 kAlevel the poloidal drift is zero, so the beam will strike the injectorafter one toroidal transit. Note that here a cylindrical system isassumed.

Referring to the diamagnetic to paramagnetic transition, once the beamhas been injected and is centered in the modified betatron, the questionthen is about the individual particle orbits in the beam. Each particlefeels an inward force due to the focusing fields and an outward forcedue to the self fields. If the latter dominates, the particle has an F×Bdrift in the counterclockwise direction, analogous to thecounterclockwise whole beam drift for an outward image force discussedabove. Then the J (poloidal) B (toroidal) force is inward. In this case,the beam is said to be diamagnetic. This is analogous to the terminologyin plasma physics, where the poloidal current is diamagnetic if it givesan inward force. On the other hand, if the focusing force dominates, theF×B drift is clockwise and the beam is said to be paramagnetic.

If the beam has uniform density and radius ρ_(b), the outward force isthe electrostatic force canceled by the magnetic force and fractionalcharge neutralization. A test charge q at ρ=ρ_(b) feels an outward force##EQU9## The inward focusing force is given by ##EQU10## so thecondition for a paramagnetic beam in a vacuum modified betatron is##EQU11## Note that a high current beam is generally diamagnetic.However as it accelerates, the left hand side becomes smaller as γincreases, and the right hand side becomes larger because B_(z) isproportional to γ. Thus for a high current beam which starts outdiamagnetic, as it accelerates it ultimately makes a transition andbecomes paramagnetic.

One might think this simply means that the poloidal rotation of theparticle stops and changes direction. Actually the situation isconsiderably more complex, and also worse from the point of view ofoperation of the modified betatron. In a recent series of papers, seeManheimer, supra; Finn, supra; J. M. Grossman, J. M. Finn and W. M.Manheimer, Phys. Fluids, to be published, it has been shown that subjectonly to the constraint that the acceleration time is very long comparedto the drift time, an approximation well-satisfied in the NRL modifiedbetatron (but not satisfied at all in particle simulations of thedevice), the diamagnetic to paramagnetic transition necessarily resultsin a change of topology of the beam. The outer beam particles firstbecome paramagnetic and in doing so scrape off the edge of the beam andform a large minor radius hollow beamlet. As the energy continues toincrease, the scrapeoff point moves inside the beam and inner beamparticles continue to add to the outside of the hollow beamlet. Theprocess is completed when the beam has turned itself completely insideout and has gone from a solid to a hollow beam.

Although this process is complicated, it is very easy to see that inmaking the transition, the beam must turn itself inside out. To showthis, it is only necessary to invoke the conservation of toroidalcanonical momentum P₇₄ . If the poloidal magnetic field is given by∇Ψ×i.sub.θ /R, then ##EQU12## To evaluate P₇₄ , note that γ=(E-qφ)/mc²where φ is the electrostatic potential. For a cylindrical beam of radiusr_(b), ##EQU13## The flux Φ has three components. First there is theflux of the vertical field itself, assumed uniform ##EQU14## Secondly,there is the flux associated with the focusing field. If the field indexis 1/2, this is ##EQU15## Note that Φ_(f) has this form both for ρ<ρ_(b)and ρ>ρ_(b). Finally, there is the flux associated with the self field,

    Φ.sub.s =(V.sub.74 /c.sup.2 Rθ(ρ)            (14)

Thus if V₇₄ ≈c, near the axis (ρ≈0) one has that ##EQU16## Since qB_(z)<O for the modified betatron, one has the result that if n is largeenough that the second term dominates (that is, if the beam isdiamagnetic), P.sub.θ (ρ=0) is a relative minimum. However far from thebeam P.sub.θ is dominated by the focusing force which have the oppositesign. Thus P.sub.θ as a function of ρ for a diamagnetic beam is shown inFIG. 3a. On the other hand, if the beam is paramagnetic, the first termon the right hand side of Eq. (15) dominates so P.sub.θ (ρ=0) is arelative maximum, and P.sub.θ (ρ) is shown in FIG. 3b.

The crucial point is that in a configuration which has θ symmetry,P.sub.θ is an exact constant of motion. Consider then the orbits at ρ= 0and ρ=ρ_(b) for a diamagnetic beam. The former is inside the latter andhas a lower value of P.sub.θ according to FIG. 1 a. After diamagnetic toparamagnetic tranistion, these values of P.sub.θ cannot change. Howevera paramagnetic beam has the reference orbit at a relative maximum sothat this must correspond to the orbit initially at ρ=ρ_(b) in thediamagnetic beam. Thus in making transition the beam must, at the veryleast, turn itself inside out.

Actually, as shown in Manheimer, supra; Finn, supra; and Grossman supra;not only does the beam turn itself inside out, it transitions from asolid to hollow beam. In doing so, the beam could strike the wall andthereby disrupt. Conditions for the beam to remain confined ontransition are given in the three references. However, even if the beamdoes not remain initially confined, it is not certain it can remainconfined long after suffering such a violent perturbation. If nothingelse, the hollow profile produced is diocotron unstable.

One other potential difficulty with the modified betatron is theresistive wall instability. If a beam of density n and radius ρ_(b) iscentered in a cylindrical tube of radius a, the frequency of aperturbation at frequency varying like exp ilφ is ##EQU17## whereV.sub.φ is the rotation frequency of the electrons ##EQU18## and ω_(b)is the frequency of rotation generated by the focusing fields, ##EQU19##The focusing fields produce a rotation in the negative direction. Sinceq<o, the rotation of the beam itself is in the positive direction forthe case of a vacuum beam, f=o. The sign of the frequency is such thatas long as ω>o, wall resistivity gives rise to growth of this mode. Thiscan be understood by noting that since the beam has only negativecharge, the net force from any perturbation must be outward. Thus, wallresistivity will cause the beam to spiral outward, corresponding toinstability. Since the natural frequency of the l=1 mode is very low,the sign of this frequency can be changed by the focusing fields,thereby stabilizing this mode. The required condition for this is thatthe beam current, as defined above, be in the low or intermediateregime. However the l=2 mode has a significantly larger frequency sothat in the vacuum modified betatron, it cannot be stabilized by thefocusing fields.

SUMMARY OF THE INVENTION

Accordingly one object of the present invention is to provide a novelmodified betatron that allows an injected beam to spiral inward towardthe center of the betatron with the bam having a net inward poloidalforce throughout the beams path.

Another object of the present invention is to provide a modifiedbetatron that does not have a diamagnetic to paramagnetic transition inthe injected beam.

Another object of the present invention is to provide a modifiedbetatron that does not have an l=2 resistive wall instability.

Yet another object of the present invention is to provide a novelmodified betatron that has the self fields inward in the poloidal plane.

Another object of the present invention is to provide a novel modifiedbetatron that has a very low density background plasma.

These and other objects of the present invention are achieved with amodified betatron for accelerating charged particles, said betatronhaving a toroidal vacuum chamber in which particle acceleration takesplace; means for generating a betatron magnetic field for acceleratingcharged particles in said vacuum chamber; means for generating a chargedparticle beam in said vacuum chamber; means for generating an electricfield to oppose the electric field induced by the diffusion of the selfmagnetic field of the beam; and means for energizing said electric fieldgenerating means for only the period during which the self magnetic fluxdiffuses out of said torodial chamber, wherein the improvement comprisesmeans for directing the self magnetic field of the beam inward in thepoloidal plane.

BRIEF DESCRIPTION OF THE DRAWINGS

a more complete appreciation of the invention and many of the attendantadvantages thereof will be readily obtained as the same becomes betterunderstood by reference to the following detailed description whenconsidered in connection with the accompanying drawings, wherein:

FIGS 1(a) and 1(b) are cross-sectional views of an offset beam in aplasma modified and vacuum modified betatron.

FIGS. 2(a) and 2(b) are cross-sectional views of a beam centered plasmamodified and vacuum modified betatron.

FIGS. 3(a) and 3(b) are graphs of the dependence of P₇₄ for adiamagnetic beam and a paramagnetic beam.

DESCRIPTION OF THE PREFERRED EMBODIMENT

Referring now to the drawings, wherein like reference numerals designateidentical or corresponding parts throughout the several views, and moreparticularly to FIG. 1A thereof, there is shown a cross-section of anoffset beam 10 located in liner 12 of the torus of a vacuum chamber 14of a plasma modified betatron. Field coil 18, which encircles the torusand coil power source 20 therefor, provides the vertical magnetic fieldfor accelerating the electron beam and the focusing fields to oppose theelectric field induced by diffusion of the self field of the beam. Theforce F on the beam 10 is in a negative or inward direction from thecenter of the liner 12. The plasma 16 in the chamber contributes ions tothe beam 10 which negate the force of the image charges (not shown) inthe wall of the vacuum vessel due to the beam 10. The image chargescontribute a force ##EQU20## the plasma contributes a force ##EQU21##and the focusing fields generated by field coil 18 of the modifiedbetatron contribute a force ##EQU22## resulting in a net inward force##EQU23## Here, Bz is the vertical magnetic field, produced by fieldcoil 18 ρ is the displacement of the beam from the equilibrium orbit inthe polodial plane, f is the distance from the beam center to the liner,q is the charge, I is the current, f is the fractional neutralization, γis the electron energy divided by mc², where m is the mass of anelectron and c is the speed of light, and Ro is the major radius of theequilibrium orbit of the beam.

FIG. 2A shows the beam 10 in diamagnetic transition after having beencentered in the torus of the vacuum chamber 14 in a plasma modifiedbetatron with field coil 18, which encircles the torus and coil powersource 20 therefor. The net force on the beam 10 is negative or inwardtoward the center of the chamber 14. The focusing force of the plasmamodified betatron is -qBzρb/2Ro, where ρb is the radius of the beam. Theimage charges in the walls of the vacuum chamber 14 due to the beam 10contribute an outward force γ⁻² qI/10ρb, and the plasma 16 in the vacuumchamber 14 contributes a force -qIf/10ρb resulting in a net force on thebeam equal to ##EQU24## This is in sharp contrast to the force on thebeam in a vacuum modified betatron during injection, see FIG. 1b, wherethe net force on the beam 10, after being injected by diode 22, ispositive or outward and away from the center of the chamber 14. Theimage charges in the wall of the chamber 14 contribute a force γ⁻²qI/10δ, and the focusing force of the vacuum modified betatroncontribute a negative or inward force -qB₂ ρ/2Ro. The net force on thebeam is then

    γ.sup.-2 qI/10δ-qB.sub.2 ρ/10Ro

and is positive or outward since the image force is larger than thefocusing forces for increasing current I. Similarly, in the diamagnetictransition of the beam in the vacuum modified betatron, the imagecharges contribute a force qIγ⁻² /10ρb and the focusing forcescontribute a force -qBzρb/2Ro. The net force is

    qIγ.sup.-2 /10ρb-qB.sub.z ρb/zRo

and is positive or outward for increasing current I.

In the plasma assisted modified betatron, a very low density fullionized preformed plasma is produced in the modified betatron. Theelectron density in this plasma should be considerably less than thebeam density; an electron density of 10⁸ cm⁻³ would be a typicaldensity. This plasma could be produced by distributing along the liner alarge number of very small, low power plasma guns. These could be firedsimultaneously, or else at some sequence predetermined so that thechamber fills nearly uniformly with plasma. For instance, the gunsfarthest from the horizontal midplane could be fired earlier. (There area number of other ways to produce the background plasma includinginjection and ionization of a gas puff, a microwave plasma discharge, alaser ionized pellet, injection of a plasma from a single plasma gun, ordisrupted tokamak plasma). Since the modified betatron has a verticalfield that is 5-10% of the toroidal field, the plasmas will drift intothe center along the field line. Since the plasma has no equilibrium, itwill of course drift out also. However, the time for the plasma to driftin and out is tens of microseconds. When the plasma has a nearly uniformdensity of say 10¹⁰ cm⁻³ the beam is fired in. Since the toroidaltransit time of the beam is about 20 nsec and the poloidal drift time isabout 200 nsec, the plasma acts as a stationary background. In responseto the electrostatic force of the beam however, plasma electrons areexpelled and plasma ions are sucked in, providing a partial chargeneutralization of the beam. This charge neutralization should beslightly more than that required to change the sign of the self forcesfrom outward to inward. Since there is a large magnetic inward selfforce, the fractional neutralization f should be larger than γ⁻² where γis the electron energy divided by mc².

Since the self forces are now inward in the poloidal plane, wallresistivity will always spiral the beam toward the center. Once the beamis centered, it can be accelerated by increasing the vertical field andflux as in a conventional betatron. In this way the acceleration of ahigh current (multi kiloamp) electron ring from an initial energy of oneto three megavolts to a final energy of tens of megavolts is achieved.

More specifically, the difficulty of beam injection at high orintermediate range currents, the problem of the diamagnetic toparamagnetic transition which occurs even well into the low currentrange, and the problem of l≧2 resistive wall instability in the modifiedbetatron all result from the fact that γ⁻² -f>0, or that the beam selfforces are outward. One possible cure for all these problems then is tooperate the high current modified betatron in the presence of a lowdensity background plasma, so that γ⁻² -f changes sign. This changes thesign of the self forces in the poloidal plane from outward to inward andprovides cohesion for the beam itself. On injection, the image forcesare now toward the center so that wall resistivity can trap the beam inthe center since if the beam is near the lines, it will always spiralinward in the poloidal plane. The beam will always be paramagnetic, sothe problems of making transition are eliminated. Also, the frequency ofthe diocotron mode will be negative so that the resistive wallinstability will be stabilized. The plasma densities required are low.For instance for a 10 kA, 2 cm radius beam with γ=4, background iondensity of order 10¹⁰ cm⁻³ is required. For a 1 kA beam, it is an orderof magnitude lower.

Although the betatron is envisioned as operating in the presence of abackground plasma, the scheme proposed here has little in common with a"plasma betatron". There, the beam is formed from runaway electrons andthe beam density is small compared to the background plasma density. SeeG. J. Budker in proceedings of CERN Symposium on High EnergyAccelerators and Pion Physics, Geneva 1956 (CERN Scientific InformationService, Geneva 1956) Vol. 1, pp. 68-76; J. G. Linhat in proceedings ofthe Fourth International Conference on Ionization Phenomena in Gases,Uppsala, Sweden, ed. N. Robert Nilsson (North Holland, Amsterdam 1960)p. 981; and H. Knoepfel, D. A. Spong and S. J. Zweben, Phys, Fluids 20,511 (1977). An alternate scheme involves injection of an astron gunproduced electron ring into a high density collisional plasma. D. P.Taggart, M. R. Parker, H. J. Hopman, R. Jayakumar and H. H. Fleischmann,Phys, Rev. Lett. 52, 1601 (1984). In the scheme proposed here, a beam isstill externally injected, and the preformed plasma denisty is lowcompared to the beam density.

The most likely approach to produce a fully ionized plasma at a densityas low as 10¹⁰ cm⁻³ would be to produce the plasma at much higherdensity and let it expand into the entire toroidal chamber. The factthat there is a vertical field which is typically five to ten percent ofthe plasma can drift along a field line from the edge of the chamber tothe center. One possible scheme would then be to distribute a largenumber of very small plasma guns along the top of the liner. These couldbe made to fire simultaneously so that plasma would line the top of thechamber. As it drifted in along the field the density would decrease dueto the expansion. The plasma would expand along the field, (and alsooutward in major radius) filling the torus. Since the expansion velocityis about 10⁶ cm/sec, it would take many microseconds for the torus tofill. However this is a very long time compared to the 20 nsec transittime of the beam in major radius. Thus when plasma conditions areoptimum, the beam would be fired in.

The system envisioned has the beam injected into a very low densityplasma in a modified betatron configuration. The question then is howdoes the plasma respond to the beam, and more specifically how is frelated to plasma, beam and system parameters. Here, the electron andion responses are treated separately. Throughout, it is assumed that thebackground plasma has sufficiently small density compared to the beam,that the electric fields from the beam dominate those from the plasma.Since the plasma is nearly at rest, there is no γ⁻² cancellation of selfelectric fields.

The plasma electrons react on two times scales, the fast inertial timescale and the slow collisional time scale. When the beam enters theplasma, a strong inward electric field is set up, E=-2πne ρiρ, where acylindrical model has been adapted for the beam and ρ is the radius (inthe poloidal plane). Assuming this field is set up slowly compared to aninverse cyclotron time (2.5×10⁻¹⁰ sec for a 2 KG field), the beamelectrons respond by E×B drifting in the θ direction and driftingoutward due to the inertial drift. It is the intertial drift ##EQU25##which expels the electrons, (note q=-|e|) from the beam. If an electronstarts out at ρ_(i) when n_(b) =t=0, Eq. (19) can be integrated once togive ##EQU26## Equation (20) has an apparent divergence, but of coursethe expression for E and ρ are valid only for ρ<ρ_(b), the beam radius.However, Eq. (20) does show that the electrons are totally expelled bythe beam if ##EQU27## For the 10 kA modified betatron parameters, ω_(be)² /2ω_(ce) ² ≧0.2, so about 20% of the plasma electrons are expelledfrom the beam region as the fields are being set up.

Now consider the longer (collisional) time scale. Electron-ioncollisions in the plasma cause a drag force which gives rise to anoutward drift ##EQU28## Thus the electron radius increases exponentiallyin time with growth rate νω_(br) ² /2ω_(ce) ². Classically ##EQU29## Fortemperatures of about 1 eV and n_(p) ≈10¹⁰ cm⁻³, the Coulomb logarithmλ=10, so ν≈3×10⁵. Thus the remaining electrons are expelled on a timescale of about 20 μsec. Since the electrons are forced away from thebeam on a 20 μsec time scale, and even without the beam, the electronscannot be confined in a toroidal chamber, the electrons are expected tobe expelled on a time scale of some tens of microseconds. This time islong compared to the time for the beam to center itself, but short onthe time scale of the beam acceleration. Thus, once the beam begins toaccelerate, there should be virtually no plasma electrons present.

For the ions however, the story is different because there is a strongattractive force between the ions and the beam. The ion response is nowconsidered.

If an ion is trapped near the center of the beam, its oscillationfrequency is ω_(i) =(2πn_(b) e² /M)^(1/2) -2×10⁸ sec⁻¹ for protons in astandard 10 kA beam. Since this is ten times larger than the ioncyclotron frequency, the ions are effectively unmagnetized. The ionoscillation time is also much less than the poloidal drift time of thebeam.

The ion is initially at rest and when the beam enters, the ion begins tooscillate due to the electric field. Since ions initially within thebeam do not leave, and other ions initially outside the beam spend atleast part of their oscillation inside the beam, the ion density in thebeam increases.

To estimate the steady state ion density, the system is assumed tobecylindrically symmetric about the beam center. Furthermore all ionsoscillate with slightly different frequencies so that after severaloscillations, the ions phase mix and are distributed uniformly alongtheir phase orbits. Then, it has been shown that the ion distributionfunction in velocity space is ##EQU30## where the constants of motionare ##EQU31## ρ(H) is the maximum radius on an ion with energy H, ω(H)is the oscillation frequency of an ion with energy H, φ(ρ) is theelectrostatic potential of the beam and n_(i) is the preformed plasmaion density. If the ion orbit is entirely within the beam ##EQU32## Ifthe ion is initially outside the beam ##EQU33## whereH(ρ_(b))=qφ(ρ_(b)). The frequemcy of the ion outside the beam cannot becomputed in closed form. As an approximation to it, use the frequency ofan ion which rotates around the beam

    ω(H)=ω.sub.i ρ.sub.b /ρ(H)             (27)

If ρ(H)>>r_(b) it is not difficult to see that this estimate is off by afactor of order [lnρ(H)/ρ_(b) ]^(1/2). The total number of ions trappedin the beam is then ##EQU34## The first integral, labeled I is justπρ_(b) ² n_(i) since it represents those ions originally in the beam.The last integral II, is estimated. To do so, set ##EQU35## where d isthe maximum radius in the poloidal plane. In this case, the integralII≈πn_(i) ρ_(b) d(1+ln d/ρ_(b))^(1/2) so that ##EQU36## Thus on an ionoscillation time scale, the ion density in the beam is significantlyenhanced. If the beam is centered as it is in steady state so d/ρ_(b)≈5, the ion density might be enhanced by nearly an order of magnitude.If the beam is near the liner, as it is on injection, a reasonable guessfor d is the distance for the liner, so d≈2ρ_(b). In this case the iondensity can be enhanced by perhaps a factor of 2.

Thus between the expulsion of plasma electrons from the beam region, andpartial trapping of plasma ions within the beam, the neutralizing iondensity within the beam should be at least as large as the initial iondensity in the plasma. Finally it should be noted that on the time scaleof beam acceleration, there should be no plasma electrons in the system.Therefore the accelerating field produced by field coil 18 will not beshielded out by any background plasma, and the beam should accelerate aswould a vacuum betatron.

Another potential problem with the plasma assisted modified betatron isthe ion resonance instability. This instability arises from the factthat the ion oscillation frequency is different from the electronrotation frequency in the poloidal plane. This is a particular concernbecause is it now established that two other similar devices, HIPAC (J.D. Daughty, J. E. Emnger and G. S. Janes, Phys. Fluids 12, 2677) andSPAC II (A. Mohri, M. Masuzaki, T. Tsuzuk, and K. Iruth, Phys. Rev. RevLett. 34, 574 (1975)) were distruped by the ion resonance instability.However in both of these devices the beam nearly filled the chamber,making it particularly susceptible to the l=1 instability. For themodified betatron with ρ_(b) <<a, the l=1 mode should not go unstableand the main danger is an l=2 mode. This mode was not observed on HIPACor SPAC II. In the modified betatron, even if paremeters are right forit, there is still a good chance that it will be stabilized by thediffuse profile.

The ion resonance instability can occur if the ion bounce frequency isroughly equal to the l=2 diocotron mode frequency. According to W. M.Manheimer, Particle Accel. 13,209 (1983), this can occur only if##EQU37## For the standard parameters of the 10 kA beam, ω_(be)≈2.4×10¹⁰, ω_(ce) 4×10¹⁰, M/m=1800, Eq. (31) above reduces to γ<3. Thusas long as γ>3 after self field diffusion, the plasma background shouldnot give rise to an ion resonance instability.

Another concern is the ion streaming instability. In this instability acyclotron mode on the beam resonates with the stationary ions. Theinstability only occurs if the parellel wave number is in the range##EQU38## and the growth rate is roughly ##EQU39## For the standardparameters here, the growth time is about 20 μsec and the range ofunstable k is several percent and is dependent on γ. The idea then is toaccelerate the beam so that γ changes by several percent in a few growthtimes.

In the modified betatron, the difficulties with injection, diamagneticto paramagnetic transition, and l=2 resistive wall instability all havetheir origin in the fact that the beam self forces are outward int hepoloidal plane. By utilizing a very small amount of fractional chargeneutralization, the sign of the self force reverses and becomes inwardin the poloidal plane. This greatly alleviates the problem of injection,since if the beam is injected near the liner, wall resistivity issufficient to move it from the liner to the center. Also there is nodiamagnetic to paramagnetic transition since beam will always be in theparamagnetic state (the same state as a conventional betatron). Alsosince the poloidal forces are inward, the =2 resistive wall instabilitywill be stabilized.

The plasma betatron or runaway tokamak betatron, on the other hand, alsohas inward self forces. However, unlike the plasma assisted modifiedbetatron there is no separately injected, well-defined, monoenergetic,centered beam. Thus extraction of the beam from one of these devicesshould be nearly impossible.

Lastly, it should be noted that the electron beam should have a densityat least 10¹⁰ electrons per centimeter cubed, with the background plasmahaving a density 1-10% of the beam density.

Obviously, numerous (additional) modifications and variations of thepresent invention are possible in light of the above teachings. It istherefore to be understood that within the scope of the appended claims,the invention may be practiced otherwise than as specifically describedherein.

What is claimed and desired to be secured by Letters Patent of theUnited States is:
 1. In a modified betatron for accelerating chargedparticles, said betatron having a toroidal vacuum chamber in whichparticle acceleration takes place; means for generating a betatronmagnetic field for accelerating charged particles in said vacuumchamber; means for generating a charged particle beam having a selfmagnetic field in said vacuum chamber; means for generating an electricfield to oppose the electric field induced by the diffusion of the selfmagnetic field of the beam; and means for energizing said electric fieldgenerating means for only the period during which the self magneticfield diffuses out of said torodial chamber; wherein the improvementcomprises means for directing the self forces of the beam inward in thepoloidal plane.
 2. An apparatus as described in claim 1 wherein thedirecting means is a background plasma having a density between 1% and10% of the beam density disposed throughout the vacuum chamber, saidbeam density being at least 10¹⁰ electrons per centimeter cubed.
 3. Anapparatus as described in claim 2 wherein the background plasma is anionized gas puff.
 4. An apparatus as described in claim 2 wherein thebackground plasma is a microwave plasma discharge.
 5. An apparatus asdescribed in claim 2 wherein the background plasma is a laser ionizedpellet.
 6. An apparatus as described in claim 2 wherein the backgroundplasma is a disrupted tokamak plasma.
 7. An apparatus as described inclaim 2 wherein the background plasma is a plasma from a single plasmagun, said plasma gun located in the vacuum chamber and firing the plasmainto the vacuum chamber.